L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.169 − 0.294i)3-s + (0.500 + 0.866i)4-s − 2.88·5-s + (0.169 + 0.294i)6-s + (1.77 + 3.06i)7-s − 3·8-s + (1.44 + 2.49i)9-s + (1.44 − 2.49i)10-s + (−0.5 + 0.866i)11-s + 0.339·12-s + (1.66 − 3.19i)13-s − 3.54·14-s + (−0.490 + 0.849i)15-s + (0.500 − 0.866i)16-s + (2.61 + 4.52i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.0981 − 0.169i)3-s + (0.250 + 0.433i)4-s − 1.28·5-s + (0.0693 + 0.120i)6-s + (0.669 + 1.16i)7-s − 1.06·8-s + (0.480 + 0.832i)9-s + (0.456 − 0.789i)10-s + (−0.150 + 0.261i)11-s + 0.0981·12-s + (0.463 − 0.886i)13-s − 0.947·14-s + (−0.126 + 0.219i)15-s + (0.125 − 0.216i)16-s + (0.633 + 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.293 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.513055 + 0.693881i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.513055 + 0.693881i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (-1.66 + 3.19i)T \) |
good | 2 | \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.169 + 0.294i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + 2.88T + 5T^{2} \) |
| 7 | \( 1 + (-1.77 - 3.06i)T + (-3.5 + 6.06i)T^{2} \) |
| 17 | \( 1 + (-2.61 - 4.52i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.11 + 5.39i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.339 - 0.588i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.21 + 7.29i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 8.40T + 31T^{2} \) |
| 37 | \( 1 + (1.76 - 3.05i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.87 + 4.97i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.660 - 1.14i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 6.10T + 47T^{2} \) |
| 53 | \( 1 + 6.08T + 53T^{2} \) |
| 59 | \( 1 + (-7.05 - 12.2i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.00 + 3.48i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.05 - 10.4i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 - 0.904T + 79T^{2} \) |
| 83 | \( 1 - 4.90T + 83T^{2} \) |
| 89 | \( 1 + (-2.82 + 4.89i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.39 + 2.42i)T + (-48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.23226553800205213676080309910, −12.24505143956365416743257102912, −11.62167552872094285268642081951, −10.46358404282945214892883711119, −8.590147049804193543294919976648, −8.177888031923234930568388056919, −7.39295259010814437373898649618, −5.95184392495624034168651774020, −4.40597357128896291037508567201, −2.68195133408264526185760946624,
1.07096009591058075893692776287, 3.46295341241537849987082325893, 4.51667750288562432253196531162, 6.48910314984411861532895926726, 7.57562392098109479669587193035, 8.730484495730862413889235310425, 9.994727333803601536656683056767, 10.81240666346102103334470044089, 11.68390435241481062027274935686, 12.33278938642137365649833878983